VOLUME 82, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 31 MAY 1999 Spin Density Waves in Thin Chromium Films A. M. N. Niklasson,* Börje Johansson, and Lars Nordström Condensed Matter Theory Group, Physics Department, Uppsala University, Box 530, S-75121 Uppsala, Sweden (Received 12 January 1999) The magnetic profile of Fe CrN Fe bcc(001) films has been calculated by means of first principles density functional theory. It is shown that the magnetic profile of the chromium spacer can be expressed in terms of spin density waves (SDW). The dispersion and amplitude of the SDW are determined and the effects from the finite film thickness are observed and discussed. It is found that the SDW wave vectors are quantized and that for certain Cr thicknesses two SDWs with different wavelengths coexist. Connections to the magnetic interlayer coupling are discussed. [S0031-9007(99)09241-8] PACS numbers: 75.30.Fv, 73.20.Dx, 75.30.Et, 75.70.Cn Chromium metal exhibits a great number of complex ing wave vector. The energy is lowered by forming the magnetic phenomena and as the archetype of a spin density SDW due to the opening of a partial band gap at the Fermi wave (SDW) it has been intensively studied [1]. The energy. However, in a layered system where the perpen- SDW in bulk chromium is generally accepted as a typical dicular symmetry is broken it is not clear that the same example of how the electronic structure and the topology mechanism is appropriate in describing the SDW stabiliza- of the Fermi surface may influence magnetism in itinerant tion. Of special interest is also the influence of the mag- systems. However, the richness of the phenomena is even netic interfaces at the boundaries of the Cr film. These more manifested in the properties of Fe Cr films where one proximity effects originating from the ferromagnetic Fe finds effects like oscillatory magnetic interlayer coupling films have recently attracted much attention. Especially [2], giant magnetoresistance (GMR) [3], and noncollinear the range of the proximity is of importance for the interfa- exchange coupling [4,5]. Thus it is of great importance cial influence on the SDW. to investigate the nature of the magnetic structure of The magnetic structure has been calculated self- chromium in layered systems, where several experiments consistently within the framework of density functional have confirmed that a SDW is formed already for relatively theory [19,20] in the local spin density approximation thin Cr films [6­10]. (LSDA) [21,22]. The calculational scheme is based on the The properties of Fe Cr systems have recently also at- linearized muffin-tin-orbital method [23] within a Green's tracted a lot of theoretical attention, not least in connec- function technique for surfaces and interfaces [24]. This tion to the magnetic interlayer exchange coupling (IXC) approach has the advantage that it can handle semi-infinite and GMR effects; see, e.g., Refs. [11­15]. However, hith- systems with a broken perpendicular translational symme- erto theoretical first principles studies have not been very try, i.e., it does not rely on a slab or supercell geometry. successful in reproducing the SDW character of the mag- The interface systems investigated in the present work netism in Cr, due to computational difficulties with the consist of chromium films embedded between two semi- large SDW unit cell size together with the weak energy infinite iron crystals. All interface calculations were done dependence on its periodicity. In fact, most studies on the at the Cr bcc lattice constant, i.e., no relaxations were IXC of Fe Cr Fe were based on RKKY-like schemes, as- taken into account. A mesh of 36 special k-points was suming the spacer layers being nonmagnetic [13,14]. For used in the irreducible part of the 2-dimensional Brillouin the same reason calculations for antiferromagnetic bulk zone (2DBZ). chromium were for a long time limited to the commen- The calculated magnetic profile MaN n of the Fe CrN surate state with two atoms per unit cell [16]. Only very Fe bcc(001) film oscillates with the layer position n within recently, calculations were successfully performed for a re- the chromium film with a period close to 2 ML as is seen alistic long wavelength bulk SDW [17]. in the example displayed in the inset of Fig. 1. This is very In the present Letter we present self-consistent first similar to what is found in bulk chromium and shows many principles electronic structure calculations of the layered characteristics of a SDW. We find that the Cr magnetic resolved spin moments in Fe CrN Fe bcc(001) films con- moments for an alignment a F, AF [ferromagnetic (F) sisting of up to 52 Cr atomic layers. In this thickness or antiferromagnetic (AF)] of the Fe layers are generally range it is possible to safely observe a full period of the well described by SDW, which in bulk chromium has about 21 monolayers X (ML) between each node. Of major interest is the mecha- MaN n AaN,i sin qaN,ipn 1 faN,i , (1) nism behind the formation of the SDW. In bulk Cr the i SDW can be ascribed to nesting between parallel sheets of i.e., as a superposition of sinusoidal oscillations with wave the paramagnetic Fermi surface [18], which gives rise to vectors qaN,i, amplitudes AaN,i, and phases faN,i. Here and a peak in the q-dependent spin susceptibility at the nest- below we choose the propagation direction of the SDW to 4544 0031-9007 99 82(22) 4544(4)$15.00 İ 1999 The American Physical Society VOLUME 82, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 31 MAY 1999 closest to the Fe interfaces. The maximal absolute error of 50 Fe/Cr /Fe bcc(001) (F) =AF, m=0 51 the moment at any inner layer is smaller than 0.05mB and =AF, m=1 0.4 the 2-norm of the fitting error kDMa =AF, m=2 ) N n k2 N , 0.01Aa N . ) 0.2 µ We may thus conclude that the magnetic profile of thin 40 =F, m=0 N =F, m=1 0 Cr films indeed can be interpreted in terms of SDW. =F, m=2 Moreover, the range of proximity effects from the Fe -0.2 30 interfaces is limited to the interface Cr atomic layer, i.e., -0.4 magnetic moment ( only the moment of this Cr layer deviates substantially from the behavior described by the SDW of the Cr film. 0 10 20 30 40 50 20 atomic Cr layer However, the SDW as a whole is very sensitive to the Cr film thickness ( boundary conditions set up by the magnetic Fe layers, as will be discussed below. 10 In Fig. 2 the extracted amplitudes AaN,i are shown for the different film thicknesses N. In the present thickness 0.9 0.95 1 1.05 1.1 range there exist three different branches of i which q (2 /a) contribute to the SDW and which correspond to mai FIG. 1. The Fourier transformed magnetic moment distribu- 0, 1, 2. From these results several interesting features can tion for different Cr film thicknesses for F (solid line) and AF be observed. (dashed line) alignment of the Fe layers. The scale is shifted as (i) The amplitude of the branch corresponding to a com- to align the maxima of the curves with the corresponding thick- mensurate antiferromagnetic SDW (ma ness. The SDW dispersion follows different branches (sym- i 0), is almost constant until about 40 ML of chromium where it decays bols) which are determined by the simple dispersion rules (see text). The inset shows the magnetic moment distribution for a rapidly. This is in good agreement with experiments which 51 ML thick F film. find commensurate SDWs in very thin Cr films [7]. When the film thickness is below 30 ML the SDW amplitude of the magnetic ordering corresponding to ma 1 is sup- be perpendicular to the (001) plane so that qaN,i is the out pressed compared to its antiferromagnetic value, at 10 ML of plane component in units of the cubic reciprocal lattice thickness almost by a factor of 2. In case of a single mono- vector. layer of Cr the moment actually vanishes in the AF case When fitted to the results of the full calculations, the dis- due to the symmetry. Altering the magnetic alignment persion, i.e., qaN,i as a function of chromium thickness N, and thereby changing the periodicity of the SDW may thus is found to follow distinct branches as indicated in Fig. 1, strongly suppress or enhance the magnetic amplitude. which also shows some Fourier amplitudes of the magneti- (ii) When the amplitude of the mai 0 branch decreases zation profiles. Within each branch we find that the bound- the branch mai 2 increases in amplitude, and there is a ary conditions vary from F to AF for every second atomic layer thickness. The appearance of these branches is a di- rect consequence of the finite size of the chromium film. Generally, for a Cr film with thickness N the magnetic 0.5 moment distribution can be expressed as a sine Fourier series [Eq. (1)] with N discrete wave vector components 0.4 qaN,i Bai N 1 1 . For a F (AF) magnetic alignment of ) the Fe layers the magnetic moment profile is even (odd) µ B with respect to mirroring through the center of the Cr 0.3 spacer layer, which leads to Bai pai 2 2fai p, where =AF, m=0 pa =AF, m=1 i is an odd (even) integer. The phase will usually depend 0.2 on the boundary conditions at the Fe interfaces, but in or- amplitude ( =AF, m=2 der to allow for a commensurate AF SDW fa =F, m=0 i has to take =F, m=1 the value p 2. Our fits always give a value close to this. 0.1 =F, m=2 Thus, instead of a continuous spectrum as in the bulk, the dispersion of the SDW is quantized in different branches given by the expression qa 0.0 N,i 1 6 ma i N 1 1 in the 15 20 25 30 35 40 45 50 vicinity of the commensurate AF ordering (q 1), where number of Cr layers (N) mai is an even integer in case of a F and odd N or a AF and even N, and corresponds to the number of nodes FIG. 2. The amplitudes of the individual branches obtained of the envelope function. This simple relation is found to by fitting the calculated moment profiles. The smaller symbols indicate the corresponding amplitudes for calculations using give an almost perfect description of the results from the the von Barth­Hedin form of LSDA functional, instead of the full calculations except for the single outermost Cr layers Vosko-Wilk-Nusair parametrization. 4545 VOLUME 82, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 31 MAY 1999 jump from one branch to another. For a finite thickness However, the SDW contribution to the magnetic interlayer range around the branch jumps, two SDWs are found to coupling can be estimated in the limit of thick films. In this coexist. Splitting of the SDW peaks have been observed case the qaN value of the Cr film will deviate only slightly in neutron scattering experiments for thin Cr layers in a from the bulk value q0 and then the SDW energy per atom Fe Cr superlattice, [8] where it however was interpreted EaN should be quadratic in qaN 2 q0. This gives a SDW in terms of an interaction between SDW in neighboring contribution to the magnetic interlayer coupling propor- Cr spacer layers rather than as due to the finite thickness tional to qAF N 2 q0 2 2 qFN 2 q0 2. Inserting the simple of one Cr layer, as is clearly the reason in our present dispersion relation from above, qaN 1 6 ma N 1 1 , calculations. This splitting is illustrated by the Fourier and including branch jumps in order to minimize the en- amplitudes of the 45-ML-thick Cr F film in Fig. 1. In ergy, it is found that the total SDW contribution to the case of finite temperatures the relative distribution between interlayer exchange coupling J can be written as different harmonics may change and even multiple SDWs may coexist. J N EAF , (2) (iii) The shift in wave vector Dqa N 2 EFN ~ 21 N11 NF N, d0 N 2 N 1 1 at the N 1 1 2 branch jump is twice the difference in the q vector between F and AF alignment, i.e., jqF 2 qAFj 1 N 1 1 . This where d0 1 2 q0 is the incommensurability of the bulk is again a direct consequence of the finite set of possible SDW, and F N, d0 is a periodic function of N with the wave vectors. period 2 d0. In the interval 0 , N 1 1 , 2 d0, (iv) Because of the restriction of possible wave vectors, the dispersion of the SDW is determined by the symmetry F N, d0 2j1 2 N 1 1 d0j 2 1 . (3) of the boundary condition rather than by the nesting of the spacer material, in contrast to bulk chromium. However, The first factor in Eq. (2) gives rise to the 2 ML oscil- in the limit of thick films Cr wants to have a SDW with lation while F N, d0 contributes with a, due to branch a wave vector as close as possible to the bulk value jumps, sawtooth shaped function, with a node each 1 d0 q0 0.95 (in units of the reciprocal lattice vector). This atomic layer. Thus the SDW contribution to the IXC os- leads to branch jumps each 20th ML, i.e., the asymptotic cillates with a short period of 2 ML superimposed by a periodicity of the branch jumps is determined by the long period of 20 ML between each node or phase slip, nesting of the spacer material. A similar effect has been and with an amplitude inversely proportional to the thick- observed within a model calculation [25]. ness of the film. This periodicity is also in full agreement (v) In Fig. 2 one can see that the amplitude is not fully with experimental findings [6]. In conclusion, it is the converged with the film thicknesses used in the calcula- strong influence of the Fe interfaces on the Cr SDW as tions. Except for the commensurate SDW, the amplitudes a whole which mediates the long range IXC. However, are monotonically increasing. This, in combination with this simple estimate of the IXC assumes only one SDW branch jumps, makes it hard to directly compare with the for each alignment. For thicker films the boundary effects experimental bulk value of 0.6mB. An extrapolation of the can be diminished by allowing for a distribution of wave amplitudes to thick films seems to lead to a too small value. vectors in the SDW, in order to better adjust to the prox- However, it is found that the amplitude is a very sensi- imity effects from the Fe interfaces. Moreover, the esti- tive quantity. This can be observed by changes in the lat- mate of the coupling energy does not include the influence tice constant and by the dependence on the specific LSDA of a thickness dependent amplitude of the magnetization. functional used in the calculations. The latter is demon- (vii) As mentioned above, the phase of the SDW is strated by comparing two different LSDA parametrizations found to be almost constant, faN,i p 2, which means [21] and [22]. The amplitude is found to increase by about that the SDW adjusts itself as to maximize the interface 20% with the von Barth­Hedin instead of the Vosko-Wilk- Cr moment. This is in striking contrast with experiments Nusair functional form as shown in Fig. 2. A similar effect where the SDW prefers to have a node at the Fe interface is found when the lattice constant is increased by 1%. [8]. This disagreement is very likely due to the imperfect (vi) With the presence of a Cr SDW the RKKY-like interfaces in the experimental samples. This leads to frus- theories [26] for the IXC assuming a nonmagnetic Cr layer trations of the Cr interface atoms, which prefer an antipar- are not valid, [15] and the short wavelength oscillation is allel alignment to the Fe moments, which is minimized by instead due to the SDW. In fact, as we will notice be- the interface node of the SDW. However, for the perfect low, the SDW introduces gaps at the nesting parts of the interface case the energy is instead optimized by an inter- nonmagnetic bulk Fermi surface, so the RKKY and SDW face SDW belly due to the enhanced Cr moments at the pictures for the short wavelength oscillations are mutually Fe interface. For very thin Cr films, where experiments exclusive. Since it is not the scope of the present study, observe a commensurate SDW and hence a SDW belly at our calculations are not brought to the accuracy needed the interface, another mechanism is responsible to escape to resolve the energy difference between the F and AF the interfacial frustration [7,27]. The SDW is found to de- alignment of the Fe layers, i.e., to obtain the IXC energy. velop a noncollinear spiral form. 4546 VOLUME 82, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 31 MAY 1999 of ordering wave vectors is possible for each thickness. For some thicknesses there are two coexisting SDWs of different wave vectors. With a simple expression for the SDW contribution to the interlayer exchange coupling, we note that the jumps between different dispersion branches well explain the observed phase slips in the coupling energy as a function of thickness. We believe that this work should stimulate some new experimental work, such as, for example, a search for a systematic shift in the neutron spectra when an AF aligned sample is aligned ferromagnetically in a magnetic field, as predicted by the present Letter, or to give confirmations of the simple quantized dispersion rules for the SDW wave vectors. The discussions with Professor H. Zabel as well as FIG. 3. The D kk, E calculated at kk 0, 0.25 for non- the financial support from the Swedish Natural Science magnetic bulk Cr (dashed line) and for spin polarized Cr em- Research Council (NFR) are thankfully acknowledged. bedded in AF Fe Cr50 Fe bcc(001) (solid line). The thin lines show the corresponding DOS. Since we have already noticed that the Cr SDW does not have a wave vector that is equal to the nesting wave *Present address: Theoretical Division, Los Alamos Na- vector q0, it is of interest to see whether the calculated tional Laboratory, Los Alamos, NM 87545. layer SDWs are stabilized by a similar mechanism as the [1] E. Fawcett, Rev. Mod. Phys. 60, 209 (1988). Fermi nesting in bulk. For instance, do pseudogaps form [2] P. Grünberg et al., Phys. Rev. Lett. 57, 2442 (1986). at the Fermi level E [3] M. N. Baibich et al., Phys. Rev. Lett. 61, 2472 (1988). F? In Fig. 3 the spectral function Ds k [4] M. Rührig et al., Phys. Status Solidi A 125, 635 (1991). k, E p21 Im Tr Gs kk, E , where Gs kk, E is the Green's function of spin s, in-plane wave vector k [5] B. Heinrich et al., Phys. Rev. B 44, 9348 (1991). k, [6] J. Unguris, R. J. Celotta, and D. T. Pierce, Phys. Rev. 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